I'm studying the adiabatic theorem and there is a derivation of a geometric phase factor which incorporates terms of the form $\langle \dot \psi_n (t) | \psi_n (t)\rangle$ where the $\psi_n(t)$ are orthonormal.
It seems clear to me that this should be zero. Differentiating the normalization condition $\langle \psi_n (t) | \psi_n (t)\rangle = 1$ (preserved with unitary evolution) causes the RHS to vanish. So zero for $\langle \dot \psi_n (t) | \psi_n (t)\rangle$ and I'm at a loss as to how to understand the geometric phase which is heavily involved with these terms.
Does anyone know where I'm going wrong and why the geometric phase is not zero simply from differentiating the normalization condition?